[petsc-users] Inertia of Hermitian Matrix

[Adam Denchfield]

You can verify there is a generalization of the inertia theorem thanks to
Ikramov. Furthermore, recall the inertia theorem is about eigenvalues; for
a Hermitian matrix, you can diagonalize it in a form that has real entries,
and that preserves the eigenvalues.

Of course, being able to apply the theorem and actually applying the
theorem/method are very different. Should be something that could be added
to MUMPS.

On Feb 22, 2018 8:39 PM, "Anthony Ruth" <Anthony.J.Ruth.12 at nd.edu> wrote:

> Hello,
>
> I am trying to diagonalize a hermitian matrix using the Eigen Problem
> Solver in SLEPc, I run into errors on calls to MatGetInertia() with complex
> hermitian matrices that I did not see with real matrices. The complex and
> real versions were done with separate PETSC_ARCH. I do not know if the
> problem is with the set up of the matrix or more generally a problem
> calculating the inertia for a complex matrix.
> The matrix is created by:
>
> ierr = MatSetType(A,MATAIJ);CHKERRQ(ierr);
> ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);CHKERRQ(ierr);
>         ierr = MatSetFromOptions(A);CHKERRQ(ierr);
> ierr = MatSetUp(A);CHKERRQ(ierr);
> ierr = MatSetOption(A,MAT_HERMITIAN,PETSC_TRUE);CHKERRQ(ierr);
> ierr = MatGetOwnershipRange(A,&first_row,&last_row);CHKERRQ(ierr);
> ierr = MatSetValues(A,m,idxm,n,idxn,data,INSERT_VALUES);CHKERRQ(ierr);
> ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
> ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
>
> For a hermitian matrix, all the eigenvalues are real, so I believe it is
> possible to calculate an inertia by looking at the signs of the diagonal
> entries. I believe if it was complex but not hermitian, the complex
> eigenvalues calculating inertia would be difficult.  Is there some problem
> with doing this through sparse iterative methods? Is there a certain place
> the matrix needs to be specified as hermitian besides upon assembly?
>
> Here is the error stack I see when running:
>
>
> Mat Object: 1 MPI processes
>   type: seqaij
> row 0: (0, 0.)  (1, 1. + 1. i) (2, 0.)  (3, 0.)  (4, 0.)
> row 1: (0, 1. - 1. i) (1, 0.)  (2, 1. + 1. i) (3, 0.)  (4, 0.)
> row 2: (0, 0.)  (1, 1. - 1. i) (2, 0.)  (3, 1. + 1. i) (4, 0.)
> row 3: (0, 0.)  (1, 0.)  (2, 1. - 1. i) (3, 0.)  (4, 1. + 1. i)
> row 4: (0, 0.)  (1, 0.)  (2, 0.)  (3, 1. - 1. i) (4, 0.)
> [0]PETSC ERROR: --------------------- Error Message
> --------------------------------------------------------------
> [0]PETSC ERROR: No support for this operation for this object type
> [0]PETSC ERROR: Mat type mumps
> [0]PETSC ERROR: See http://www.mcs.anl.gov/petsc/documentation/faq.html
> for trouble shooting.
> [0]PETSC ERROR: Petsc Release Version 3.8.2, Nov, 09, 2017
> [0]PETSC ERROR: Configure options --download-metis --download-mumps
> --download-parmetis --download-scalapack --with-scalar-type=complex
> [0]PETSC ERROR: #1 MatGetInertia() line 8416 in
> /home/anthony/DFTB+SIPs/petsc-3.8.2/src/mat/interface/matrix.c
> [0]PETSC ERROR: #2 EPSSliceGetInertia() line 333 in
> /home/anthony/DFTB+SIPs/slepc-3.8.1/src/eps/impls/krylov/
> krylovschur/ks-slice.c
> [0]PETSC ERROR: #3 EPSSetUp_KrylovSchur_Slice() line 459 in
> /home/anthony/DFTB+SIPs/slepc-3.8.1/src/eps/impls/krylov/
> krylovschur/ks-slice.c
> [0]PETSC ERROR: #4 EPSSetUp_KrylovSchur() line 146 in
> /home/anthony/DFTB+SIPs/slepc-3.8.1/src/eps/impls/krylov/
> krylovschur/krylovschur.c
> [0]PETSC ERROR: #5 EPSSetUp() line 165 in /home/anthony/DFTB+SIPs/slepc-
> 3.8.1/src/eps/interface/epssetup.c
> [0]PETSC ERROR: #6 EPSSliceGetEPS() line 298 in
> /home/anthony/DFTB+SIPs/slepc-3.8.1/src/eps/impls/krylov/
> krylovschur/ks-slice.c
> [0]PETSC ERROR: #7 EPSSetUp_KrylovSchur_Slice() line 408 in
> /home/anthony/DFTB+SIPs/slepc-3.8.1/src/eps/impls/krylov/
> krylovschur/ks-slice.c
> [0]PETSC ERROR: #8 EPSSetUp_KrylovSchur() line 146 in
> /home/anthony/DFTB+SIPs/slepc-3.8.1/src/eps/impls/krylov/
> krylovschur/krylovschur.c
> [0]PETSC ERROR: #9 EPSSetUp() line 165 in /home/anthony/DFTB+SIPs/slepc-
> 3.8.1/src/eps/interface/epssetup.c
> [0]PETSC ERROR: #10 SIPSolve() line 195 in /home/anthony/DFTB+SIPs/dftb-
> eig15/sips.c
> [0]PETSC ERROR: --------------------- Error Message
> --------------------------------------------------------------
> [0]PETSC ERROR: Invalid argument
> [0]PETSC ERROR: Wrong type of object: Parameter # 1
> [0]PETSC ERROR: See http://www.mcs.anl.gov/petsc/documentation/faq.html
> for trouble shooting.
> [0]PETSC ERROR: Petsc Release Version 3.8.2, Nov, 09, 2017
> [0]PETSC ERROR: Configure options --download-metis --download-mumps
> --download-parmetis --download-scalapack --with-scalar-type=complex
> [0]PETSC ERROR: #11 EPSGetConverged() line 257 in
> /home/anthony/DFTB+SIPs/slepc-3.8.1/src/eps/interface/epssolve.c
> [0]PETSC ERROR: #12 squareFromEPS() line 131 in
> /home/anthony/DFTB+SIPs/dftb-eig15/sips_square.c
>
>
>
> regards,
> Anthony Ruth
> Condensed Matter Theory
> University of Notre Dame
>
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